The Optimal Experimental Design for Exponentiated Frech’et Lifetime Products
Abstract
:1. Introduction
2. Introduction to the Testing Procedure for the Lifetime Performance Index and Minimum Sample Size
3. Reliability Sampling Design
3.1. The Determination of the Optimal m and n When the Termination Time T Is Fixed
- Inspection cost CI: the cost of using the inspection equipment for each inspection;
- Sample cost Cs: the cost for one test unit in the sample;
- Operation cost Co: the cost per unit of time, encompassing expenses like personnel costs and the depreciation of test equipment;
- Installation cost Ca: the fixed cost for installing all test units.
- Step 1: Provide the predetermined values of c0, c1, , , , T, L, and the costs of CI = a Ca, Cs = b Ca, Co = c Ca.
- Step 2: Compute and
- Step 3: Set m = 1.
- Step 4: Calculate the sample size n using Equation (11), followed by determining the associated total cost TC(m,n) by the use of Equation (12).
- Step 5: If , then m = m + 1 and go to Step 4; otherwise, go to Step 6.
- Step 6: The optimal value of m denoted by m* is found to be the minimum value of m such that TC* = TC(m,n) is attained and the related sample size n* can be obtained by using Equation (11).
- Step 7: The critical value in the critical region can be calculated as .
3.2. The Determination of the Optimal m, t and n When the Termination Time T Is Varying
- Step 1: Provide the predetermined values of c0, c1, , , , L, and the costs of CI = a Ca, Cs = b Ca, Co = c Ca.
- Step 2: Compute and
- Step 3: Set m = 1.
- Step 4: The optimal value of t* is determined to minimize the total cost TC(m,t,n) given in Equation (13). Calculate the sample size n using Equation (11) and then compute the related total cost TC(m,t*,n) by using Equation (13).
- Step 5: If , then m = m + 1 and go to Step 4; otherwise, go to Step 6.
- Step 6: We determine the optimal choice of m denoted by m* as the minimum value of m such that TC** = TC(m,t*,n) is reached and then the corresponding sample size n* is determined by using Equation (11).
- Step 7: The critical value can be calculated as .
3.3. Example
- Step 1: Take a random sample 36 in size with m* = 2 from the data set. Collect the progressive type I interval censored sample (7, 24) at the pre-set times (0.4, 0.8) with censoring schemes of 0, 5).
- Step 2: Calculate the maximum likelihood estimator for as = 8.9934. We can find the maximum likelihood estimator for c as = 1 − 8.9934 (0.00255) = 0.9907.
- Step 3: For the level of = 0.01 test, the critical value is found to be = 0.8707.
- Step 4: Since 0.9907 > = 0.8707, we can infer that the null hypothesis should be rejected and conclude that the lifetime performance index attains the required target level c0, and we claim that the production process is capable.
- Step 1: Take a random sample 43 in size from the data set. Observe the progressive type I interval censored sample = (0, 2, 4) at the pre-set times = (0.15, 0.30, 0.45) with censoring schemes of = (5, 4, 28).
- Step 2: Calculate the maximum likelihood estimator for as = 13.3962. Then, we can find the maximum likelihood estimator for as = 1 − 13.3962 (0.00255) = 0.9658.
- Step 3: For the level of the = 0.01 test, the critical value is found to be 0.8541.
- Step 4: Since 0.9658 > 0.8541, we arrive at the same conclusion to substantiate the alternative hypothesis.
4. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
0.025 | 0.05 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
p | ||||||||||
0.01 | 0.20 | 0.050 | 20 | 1629 | 1650.8 | 0.7667 | 11 | 386 | 398.8 | 0.7847 |
0.075 | 18 | 1654 | 1673.8 | 0.7667 | 10 | 391 | 402.8 | 0.7847 | ||
0.100 | 17 | 1673 | 1691.8 | 0.7667 | 9 | 395 | 405.8 | 0.7847 | ||
0.15 | 0.050 | 15 | 1371 | 1387.8 | 0.7678 | 9 | 328 | 338.8 | 0.7867 | |
0.075 | 12 | 1398 | 1411.8 | 0.7678 | 8 | 333 | 342.8 | 0.7867 | ||
0.100 | 10 | 1419 | 1430.8 | 0.7678 | 8 | 336 | 345.8 | 0.7867 | ||
0.10 | 0.050 | 10 | 1202 | 1213.8 | 0.7688 | 8 | 288 | 297.8 | 0.7886 | |
0.075 | 9 | 1224 | 1234.8 | 0.7688 | 7 | 293 | 301.8 | 0.7886 | ||
0.100 | 8 | 1243 | 1252.8 | 0.7688 | 6 | 298 | 305.8 | 0.7886 | ||
0.05 | 0.20 | 0.050 | 18 | 1056 | 1075.8 | 0.7647 | 9 | 247 | 257.8 | 0.7808 |
0.075 | 16 | 1072 | 1089.8 | 0.7647 | 8 | 250 | 259.8 | 0.7808 | ||
0.100 | 15 | 1084 | 1100.8 | 0.7647 | 7 | 253 | 261.8 | 0.7808 | ||
0.15 | 0.050 | 13 | 861 | 875.8 | 0.7659 | 8 | 203 | 212.8 | 0.7831 | |
0.075 | 11 | 877 | 889.8 | 0.7659 | 7 | 206 | 214.8 | 0.7831 | ||
0.100 | 9 | 890 | 900.8 | 0.7659 | 6 | 209 | 216.8 | 0.7831 | ||
0.10 | 0.050 | 9 | 732 | 742.8 | 0.7671 | 6 | 175 | 182.8 | 0.7852 | |
0.075 | 8 | 745 | 754.8 | 0.7671 | 5 | 178 | 184.8 | 0.7853 | ||
0.100 | 7 | 756 | 764.8 | 0.7671 | 6 | 178 | 185.8 | 0.7853 | ||
0.10 | 0.20 | 0.050 | 16 | 802 | 819.8 | 0.7632 | 6 | 188 | 195.8 | 0.7778 |
0.075 | 14 | 814 | 829.8 | 0.7632 | 6 | 189 | 196.8 | 0.7778 | ||
0.100 | 13 | 823 | 837.8 | 0.7632 | 6 | 190 | 197.8 | 0.7778 | ||
0.15 | 0.050 | 12 | 637 | 650.8 | 0.7644 | 6 | 150 | 157.8 | 0.7803 | |
0.075 | 10 | 649 | 660.8 | 0.7644 | 5 | 153 | 159.8 | 0.7803 | ||
0.100 | 9 | 658 | 668.8 | 0.7644 | 5 | 154 | 160.8 | 0.7802 | ||
0.10 | 0.050 | 9 | 528 | 538.8 | 0.7657 | 5 | 126 | 132.8 | 0.7826 | |
0.075 | 7 | 539 | 547.8 | 0.7657 | 5 | 127 | 133.8 | 0.7826 | ||
0.100 | 7 | 546 | 554.8 | 0.7657 | 5 | 128 | 134.8 | 0.7826 |
0.125 | 0.15 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
p | ||||||||||
0.01 | 0.20 | 0.050 | 3 | 54 | 58.8 | 0.8468 | 2 | 36 | 39.8 | 0.8707 |
0.075 | 3 | 54 | 58.8 | 0.8469 | 2 | 36 | 39.8 | 0.8708 | ||
0.100 | 3 | 54 | 58.8 | 0.8470 | 2 | 36 | 39.8 | 0.8708 | ||
0.15 | 0.050 | 4 | 46 | 51.8 | 0.8509 | 3 | 31 | 35.8 | 0.8747 | |
0.075 | 3 | 48 | 52.8 | 0.8503 | 3 | 31 | 35.8 | 0.8748 | ||
0.100 | 3 | 48 | 52.8 | 0.8505 | 2 | 33 | 36.8 | 0.8739 | ||
0.10 | 0.050 | 3 | 43 | 47.8 | 0.8537 | 3 | 28 | 32.8 | 0.8785 | |
0.075 | 3 | 43 | 47.8 | 0.8539 | 2 | 30 | 33.8 | 0.8778 | ||
0.100 | 3 | 43 | 47.8 | 0.8541 | 2 | 30 | 33.8 | 0.8779 | ||
0.05 | 0.20 | 0.050 | 2 | 33 | 36.8 | 0.8392 | 2 | 21 | 24.8 | 0.8618 |
0.075 | 2 | 33 | 36.8 | 0.8392 | 2 | 21 | 24.8 | 0.8618 | ||
0.100 | 2 | 33 | 36.8 | 0.8392 | 2 | 21 | 24.8 | 0.8619 | ||
0.15 | 0.050 | 2 | 29 | 32.8 | 0.8434 | 1 | 20 | 22.8 | 0.8684 | |
0.075 | 2 | 29 | 32.8 | 0.8434 | 1 | 20 | 22.8 | 0.8684 | ||
0.100 | 2 | 29 | 32.8 | 0.8435 | 1 | 20 | 22.8 | 0.8684 | ||
0.10 | 0.050 | 2 | 26 | 29.8 | 0.8470 | 2 | 17 | 20.8 | 0.8700 | |
0.075 | 2 | 26 | 29.8 | 0.8471 | 2 | 17 | 20.8 | 0.8701 | ||
0.100 | 2 | 26 | 29.8 | 0.8472 | 2 | 17 | 20.8 | 0.8702 | ||
0.10 | 0.20 | 0.050 | 1 | 25 | 27.8 | 0.8328 | 1 | 16 | 18.8 | 0.8535 |
0.075 | 1 | 25 | 27.8 | 0.8328 | 1 | 16 | 18.8 | 0.8535 | ||
0.100 | 1 | 25 | 27.8 | 0.8328 | 1 | 16 | 18.8 | 0.8535 | ||
0.15 | 0.050 | 2 | 20 | 23.8 | 0.8376 | 1 | 14 | 16.8 | 0.8602 | |
0.075 | 2 | 20 | 23.8 | 0.8376 | 1 | 14 | 16.8 | 0.8602 | ||
0.100 | 2 | 20 | 23.8 | 0.8377 | 1 | 14 | 16.8 | 0.8602 | ||
0.10 | 0.050 | 2 | 18 | 21.8 | 0.8408 | 1 | 13 | 15.8 | 0.8640 | |
0.075 | 2 | 18 | 21.8 | 0.8409 | 1 | 13 | 15.8 | 0.8640 | ||
0.100 | 2 | 18 | 21.8 | 0.8410 | 1 | 13 | 15.8 | 0.8640 |
0.025 | 0.05 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
p | ||||||||||||
0.01 | 0.20 | 0.050 | 20 | 0.03 | 1628 | 1649.6 | 0.7667 | 11 | 0.04 | 385 | 397.4 | 0.7847 |
0.075 | 18 | 0.04 | 1653 | 1672.6 | 0.7667 | 9 | 0.04 | 391 | 401.3 | 0.7847 | ||
0.100 | 14 | 0.04 | 1676 | 1691.5 | 0.7667 | 8 | 0.04 | 395 | 404.3 | 0.7848 | ||
0.15 | 0.050 | 20 | 0.07 | 1358 | 1380.3 | 0.7679 | 10 | 0.07 | 327 | 338.7 | 0.7867 | |
0.075 | 17 | 0.08 | 1386 | 1405.3 | 0.7679 | 9 | 0.08 | 332 | 342.7 | 0.7867 | ||
0.100 | 13 | 0.09 | 1411 | 1426.1 | 0.7678 | 8 | 0.10 | 336 | 345.8 | 0.7867 | ||
0.10 | 0.050 | 19 | 0.11 | 1169 | 1191.0 | 0.7691 | 9 | 0.11 | 285 | 296.0 | 0.7887 | |
0.075 | 15 | 0.12 | 1198 | 1215.8 | 0.7691 | 8 | 0.13 | 290 | 300.1 | 0.7887 | ||
0.100 | 13 | 0.13 | 1220 | 1235.7 | 0.7690 | 7 | 0.14 | 295 | 304.0 | 0.7887 | ||
0.05 | 0.20 | 0.050 | 17 | 0.03 | 1057 | 1075.5 | 0.7647 | 7 | 0.04 | 249 | 257.3 | 0.7809 |
0.075 | 14 | 0.03 | 1074 | 1089.5 | 0.7647 | 7 | 0.03 | 251 | 259.2 | 0.7809 | ||
0.100 | 12 | 0.04 | 1087 | 1100.4 | 0.7647 | 6 | 0.05 | 253 | 260.3 | 0.7809 | ||
0.15 | 0.050 | 17 | 0.07 | 853 | 872.2 | 0.7659 | 8 | 0.09 | 203 | 212.7 | 0.7831 | |
0.075 | 13 | 0.08 | 872 | 887.0 | 0.7659 | 6 | 0.08 | 208 | 215.5 | 0.7831 | ||
0.100 | 13 | 0.09 | 883 | 898.2 | 0.7659 | 7 | 0.10 | 208 | 216.7 | 0.7831 | ||
0.10 | 0.050 | 15 | 0.11 | 714 | 731.6 | 0.7672 | 7 | 0.12 | 173 | 181.8 | 0.7853 | |
0.075 | 12 | 0.12 | 731 | 745.5 | 0.7672 | 6 | 0.14 | 176 | 183.8 | 0.7853 | ||
0.100 | 11 | 0.14 | 743 | 756.6 | 0.7672 | 6 | 0.13 | 178 | 185.8 | 0.7853 | ||
0.10 | 0.20 | 0.050 | 15 | 0.03 | 803 | 819.4 | 0.7632 | 6 | 0.05 | 187 | 194.3 | 0.7779 |
0.075 | 14 | 0.04 | 813 | 828.6 | 0.7632 | 6 | 0.05 | 188 | 195.3 | 0.7779 | ||
0.100 | 12 | 0.04 | 823 | 836.5 | 0.7632 | 6 | 0.06 | 189 | 196.3 | 0.7779 | ||
0.15 | 0.050 | 14 | 0.07 | 633 | 649.0 | 0.7644 | 7 | 0.09 | 149 | 157.6 | 0.7802 | |
0.075 | 12 | 0.08 | 645 | 659.0 | 0.7644 | 5 | 0.09 | 153 | 159.5 | 0.7803 | ||
0.100 | 11 | 0.09 | 654 | 667.0 | 0.7644 | 5 | 0.09 | 154 | 160.5 | 0.7802 | ||
0.10 | 0.050 | 13 | 0.11 | 517 | 532.4 | 0.7658 | 6 | 0.15 | 124 | 131.9 | 0.7826 | |
0.075 | 12 | 0.12 | 527 | 541.5 | 0.7658 | 5 | 0.13 | 127 | 133.7 | 0.7826 | ||
0.100 | 10 | 0.13 | 537 | 549.3 | 0.7658 | 5 | 0.14 | 128 | 134.7 | 0.7826 |
0.125 | 0.15 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
p | ||||||||||||
0.01 | 0.20 | 0.050 | 3 | 0.06 | 53 | 57.2 | 0.8477 | 2 | 0.03 | 36 | 39.1 | 0.8707 |
0.075 | 2 | 0.04 | 55 | 58.1 | 0.8477 | 2 | 0.03 | 36 | 39.1 | 0.8708 | ||
0.100 | 2 | 0.04 | 55 | 58.1 | 0.8478 | 2 | 0.03 | 36 | 39.1 | 0.8708 | ||
0.15 | 0.050 | 3 | 0.12 | 47 | 51.4 | 0.8512 | 2 | 0.11 | 32 | 35.2 | 0.8757 | |
0.075 | 3 | 0.07 | 48 | 52.2 | 0.8503 | 2 | 0.12 | 32 | 35.2 | 0.8758 | ||
0.100 | 3 | 0.08 | 48 | 52.2 | 0.8505 | 2 | 0.13 | 32 | 35.3 | 0.8759 | ||
0.10 | 0.050 | 3 | 0.12 | 43 | 47.4 | 0.8537 | 3 | 0.18 | 28 | 32.5 | 0.8785 | |
0.075 | 3 | 0.13 | 43 | 47.4 | 0.8539 | 2 | 0.13 | 30 | 33.3 | 0.8778 | ||
0.100 | 3 | 0.15 | 43 | 47.5 | 0.8541 | 2 | 0.13 | 30 | 33.3 | 0.8779 | ||
0.05 | 0.20 | 0.050 | 2 | 0.04 | 33 | 36.1 | 0.8392 | 1 | 0.06 | 22 | 24.1 | 0.8632 |
0.075 | 2 | 0.05 | 33 | 36.1 | 0.8392 | 1 | 0.06 | 22 | 24.1 | 0.8632 | ||
0.100 | 2 | 0.05 | 33 | 36.1 | 0.8392 | 1 | 0.06 | 22 | 24.1 | 0.8632 | ||
0.15 | 0.050 | 2 | 0.09 | 29 | 32.2 | 0.8434 | 1 | 0.12 | 20 | 22.1 | 0.8684 | |
0.075 | 2 | 0.09 | 29 | 32.2 | 0.8434 | 1 | 0.12 | 20 | 22.1 | 0.8684 | ||
0.100 | 2 | 0.10 | 29 | 32.2 | 0.8435 | 1 | 0.12 | 20 | 22.1 | 0.8684 | ||
0.10 | 0.050 | 2 | 0.12 | 26 | 29.2 | 0.8470 | 2 | 0.12 | 17 | 20.2 | 0.8700 | |
0.075 | 2 | 0.13 | 26 | 29.3 | 0.8471 | 2 | 0.12 | 17 | 20.2 | 0.8701 | ||
0.100 | 2 | 0.13 | 26 | 29.3 | 0.8472 | 2 | 0.13 | 17 | 20.3 | 0.8702 | ||
0.10 | 0.20 | 0.050 | 1 | 0.05 | 25 | 27.0 | 0.8328 | 1 | 0.03 | 16 | 18.0 | 0.8535 |
0.075 | 1 | 0.05 | 25 | 27.0 | 0.8328 | 1 | 0.03 | 16 | 18.0 | 0.8535 | ||
0.100 | 1 | 0.05 | 25 | 27.0 | 0.8328 | 1 | 0.03 | 16 | 18.0 | 0.8535 | ||
0.15 | 0.050 | 2 | 0.12 | 20 | 23.2 | 0.8376 | 1 | 0.10 | 14 | 16.1 | 0.8602 | |
0.075 | 2 | 0.13 | 20 | 23.3 | 0.8376 | 1 | 0.10 | 14 | 16.1 | 0.8602 | ||
0.100 | 2 | 0.14 | 20 | 23.3 | 0.8377 | 1 | 0.10 | 14 | 16.1 | 0.8602 | ||
0.10 | 0.050 | 2 | 0.12 | 18 | 21.2 | 0.8408 | 1 | 0.13 | 13 | 15.1 | 0.8640 | |
0.075 | 2 | 0.12 | 18 | 21.2 | 0.8409 | 1 | 0.13 | 13 | 15.1 | 0.8640 | ||
0.100 | 2 | 0.12 | 18 | 21.2 | 0.8410 | 1 | 0.13 | 13 | 15.1 | 0.8640 |
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Wu, S.-F. The Optimal Experimental Design for Exponentiated Frech’et Lifetime Products. Symmetry 2024, 16, 1132. https://doi.org/10.3390/sym16091132
Wu S-F. The Optimal Experimental Design for Exponentiated Frech’et Lifetime Products. Symmetry. 2024; 16(9):1132. https://doi.org/10.3390/sym16091132
Chicago/Turabian StyleWu, Shu-Fei. 2024. "The Optimal Experimental Design for Exponentiated Frech’et Lifetime Products" Symmetry 16, no. 9: 1132. https://doi.org/10.3390/sym16091132